Find the domain of the function.

f(x)=2
√
x−2
−4

Write your answer in interval notation. Use inf for ∞.

Question

Find the domain of the function.

f(x)=2
√
x−2
−4

Write your answer in interval notation.  Use inf for ∞.

BlackTom AI · Accepted Answer

Let's walk through how to determine the domain of the given function, paying close attention to where the expression is defined.
First, identify the parts of the function that could cause issues: the denominator involves a square root, so we must ensure the radicand is nonnegative and that the denominator itself is not zero.
Next, examine the square root: inside the radical we have x minus 2. For a square root to be real, we require x - 2 ≥ 0, which means x ≥ 2. However, because this square root appears in the denominator, we must also ensure the denominator is not zero.
Evaluate when the denominator would be zero: the denominator is sqrt(x - 2). This is zero precisely when x - 2 = 0, i.e., x = 2. At x = 2, the whole fraction would be undefined due to division by zero.
Combine the conditions: to keep the expression defined, we need x - 2 > 0, which translates to x > 2. This automatically satisfies the nonnegativity requirement for the radicand while avoiding division by zero.
Consider any other potential restrictions: there are no other x-terms in the numerator that would impose limits, and the constant 2 and the constant subtraction of 4 pose no extra domain constraints beyond what the denominator imposes.
Thus, the set of x values for which the function is defined consists of all numbers strictly greater than 2. If you were to express this in interval notation, you would describe the domain as all x with x > 2, using the appropriate endpoint convention and infinity notation.
In summary, the domain is determined by ensuring the radicand is nonnegative and the denominator is not zero, which leads to the constraint x > 2. This excludes x = 2 and any smaller x values since those would either make the square root undefined or make the denominator vanish.

MATH136-12P-IO-202510-I-81X M04: Midterm Exam Review Quiz

Find the domain of the function. f(x)=2 √ x−2 −4 Write your answer in interval notation. Use inf for ∞.

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